Abstract:

The thesis examines a generalised problem of optimal control of a firm through
reinsurance, dividend policy and convex risk minimisation in the presence of
market friction. The major mathematical tool applied is the theory of stochastic
control for jump-diffusions. In the absence of intervention the financial reserves of
the firm are assumed to evolve according to a stochastic differential equation with
a jump component. In the second and third chapters, the objective is to derive
reinsurance and dividend policies that maximize the expected total discounted
value of a spectrally negative process in incomplete markets. The assumption
is that transaction costs are incurred whenever dividends are paid out. Several
verification theorems are derived and proved for combined singular and impulse
control. The verification theorems are new results which provide a federative
approach to the analysis of control problems involving transaction costs in finance
and insurance.
Two methodologies are examined for risk minimisation. First, we investigate risk
minimisation using zero-sum stochastic differential game theory in the presence of
transaction costs. Our major contribution in this direction is that we have investigated,
for the first time in the literature, a singular control problem for jump
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diffusion stochastic differential games. Hamilton-Jacobi-Bellman-Isaacs variational
inequalities (HJBIVI) are formulated and proved for the case of zero-sum
stochastic differential games. The notion of HJBIVI is later on extended to the
more general case of Nash equilibrium. Minimisation of risk is also studied using
g-expectation. In this case a five step scheme is formulated. The scheme constitutes
a mechanism for solving forward-backward stochastic differential equations.
The solution provided by such a scheme minimises risk of terminal wealth of an
insurance company. An existence and uniqueness theorem for the solution is provided.
Several examples are discussed, throughout the thesis, to illustrate the
theory.